Let $\mathbf{a}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\mathbf{b}$ be two perpendicular vectors in the $X O Y$-plane. A vector $\mathbf{c}$ in the same plane and having projections 1 and 2 respectively on $\mathbf{a}$ and $\mathbf{b}$ is

If $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{b}=-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are two vectors, then the vector of magnitude 28 units in the direction of the vector $\mathbf{a}-\mathbf{b}$ is

If $\bar{a}$ is a unit vector, then

$$ |\mathbf{a} \times \hat{\mathbf{i}}|^2+|\mathbf{a} \times \hat{\mathbf{j}}|^2+|\mathbf{a} \times \hat{\mathbf{k}}|^2= $$

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{b}=-2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{c}=5 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{d}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ are four vectors, then $(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=$